But our closest star is much further away and it’s an interesting challenge to send a powerful enough transmitter that far - power being the main problem- and have an antenna on Earth or locally in space that can receive the data.
There’s also a question of bandwidth. Voyagers downlink is 160 bits per second. Webb is about 25Mbps. What bandwidth is achievable at that range?
It’s doubtful the probe would be able to receive because you wouldn’t want to send a massive antenna that far out.
There’s propagation delay which would be 4.2 years in one direction in the case of our closest star Proxima Centauri.
If we take a long view - as in, the original team would be long dead when the mission completes - I suspect this is achievable.
Check out the Lucy mission which is impressive in its orbital complexity and long time horizon if you’re into this stuff.